Question: Factor the following expression: $-7$ $x^2+$ $16$ $x+$ $15$
Solution: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-7)}{(15)} &=& -105 \\ {a} + {b} &=& & & {16} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-105$ and add them together. Remember, since $-105$ is negative, one of the factors must be negative. The factors that add up to ${16}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-5}$ and ${b}$ is ${21}$ $ \begin{eqnarray} {ab} &=& ({-5})({21}) &=& -105 \\ {a} + {b} &=& {-5} + {21} &=& 16 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {-7}x^2 {-5}x +{21}x +{15} $ Group the terms so that there is a common factor in each group: $ ({-7}x^2 {-5}x) + ({21}x +{15}) $ Factor out the common factors: $ x(-7x - 5) - 3(-7x - 5) $ Notice how $(-7x - 5)$ has become a common factor. Factor this out to find the answer. $(-7x - 5)(x - 3)$